Linear Operators

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Linear Operators

Almost all operators encountered in quantum mechanics are linear operators. A linear operator is an operator which satisfies the following two conditions:

$\displaystyle \hat{A} (f + g)$	=	$\displaystyle \hat{A} f + \hat{A} g$	(43)
$\displaystyle \hat{A} (c f)$	=	$\displaystyle c \hat{A} f$	(44)

where c is a constant and f and g are functions. As an example, consider the operators d/dx and ()². We can see that d/dx is a linear operator because

(d/dx)[f(x) + g(x)]	=	(d/dx)f(x) + (d/dx)g(x)	(45)
(d/dx)[c f(x)]	=	$\displaystyle c \; (d/dx) f(x)$	(46)

However, ()² is not a linear operator because

$\begin{displaymath}(f(x) + g(x))^2 \neq (f(x))^2 + (g(x))^2 \end{displaymath}$

(47)

The only other category of operators relevant to quantum mechanics is the set of antilinear operators, for which

$\begin{displaymath}\hat{A} (\lambda f + \mu g) = \lambda^{*} \hat{A} f + \mu^{*} \hat{A} g \end{displaymath}$

(48)

Time-reversal operators are antilinear (cf. Merzbacher [2], section 16-11).

Next: Eigenfunctions and Eigenvalues Up: Operators Previous: Basic Properties of Operators